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Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Let \(G=\mathrm{GL}_{2n}\) over a totally real number field F and \(n\ge 2\). Let \(\Pi \) be a cuspidal automorphic representation of \(G(\mathbb {A})\), which is cohomological and a functorial lift from SO\((2n+1)\). The latter condition can be equivalently reformulated that the exterior square L-function of \(\Pi \) has a pole at \(s=1\). In this paper, we prove a rationality...